Algebra Examples

$\frac{(3 x^{4} + 2 x^{3} - x^{2} + 2 x - 14)}{x + 2}$

Step 1

To calculate the remainder, first divide the polynomials.

Tap for more steps...

Step 1.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

Step 1.2

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

Step 1.3

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} + 6 x^{3}$

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

Step 1.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

Step 1.6

Pull the next terms from the original dividend down into the current dividend.

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

Step 1.7

Divide the highest order term in the dividend $- 4 x^{3}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$4 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

Step 1.8

Multiply the new quotient term by the divisor.

				$3 x^{3}$	-	$4 x^{2}$
$x$	+	$2$		$3 x^{4}$	+	$2 x^{3}$	-	$x^{2}$	+	$2 x$	-	$14$
			-	$3 x^{4}$	-	$6 x^{3}$
					-	$4 x^{3}$	-	$x^{2}$
					-	$4 x^{3}$	-	$8 x^{2}$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{3} - 8 x^{2}$

$3 x^{3}$

$4 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$8 x^{2}$

Step 1.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x^{3}$

$4 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

Step 1.11

Pull the next terms from the original dividend down into the current dividend.

$3 x^{3}$

$4 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

Step 1.12

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$4 x^{2}$

$7 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

Step 1.13

Multiply the new quotient term by the divisor.

$3 x^{3}$

$4 x^{2}$

$7 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

$7 x^{2}$

$14 x$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} + 14 x$

$3 x^{3}$

$4 x^{2}$

$7 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

$7 x^{2}$

$14 x$

Step 1.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x^{3}$

$4 x^{2}$

$7 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

$7 x^{2}$

$14 x$

$12 x$

Step 1.16

Pull the next terms from the original dividend down into the current dividend.

$3 x^{3}$

$4 x^{2}$

$7 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

$7 x^{2}$

$14 x$

$12 x$

$14$

Step 1.17

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $x$ .

$3 x^{3}$

$4 x^{2}$

$7 x$

$12$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

$7 x^{2}$

$14 x$

$12 x$

$14$

Step 1.18

Multiply the new quotient term by the divisor.

				$3 x^{3}$	-	$4 x^{2}$	+	$7 x$	-	$12$
$x$	+	$2$		$3 x^{4}$	+	$2 x^{3}$	-	$x^{2}$	+	$2 x$	-	$14$
			-	$3 x^{4}$	-	$6 x^{3}$
					-	$4 x^{3}$	-	$x^{2}$
					+	$4 x^{3}$	+	$8 x^{2}$
							+	$7 x^{2}$	+	$2 x$
							-	$7 x^{2}$	-	$14 x$
									-	$12 x$	-	$14$
									-	$12 x$	-	$24$

Step 1.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x - 24$

$3 x^{3}$

$4 x^{2}$

$7 x$

$12$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

$7 x^{2}$

$14 x$

$12 x$

$14$

$12 x$

$24$

Step 1.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x^{3}$

$4 x^{2}$

$7 x$

$12$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$x^{2}$

$2 x$

$14$

$3 x^{4}$

$6 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

$7 x^{2}$

$14 x$

$12 x$

$14$

$12 x$

$24$

$10$

Step 1.21

The final answer is the quotient plus the remainder over the divisor.

$3 x^{3} - 4 x^{2} + 7 x - 12 + \frac{10}{x + 2}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$10$